Cohomology of groups and splendid equivalences of derived categories (Q2758157)

Recently [see, e.g., AMA, Algebra Montp. Announc. 2000, No. 1, Paper No. 3 (2000; Zbl 0981.20002)], the author has started to study group cohomology from a new point of view. The new structure comes from certain derived equivalences acting on group cohomology. In an earlier paper [Arch. Math. 79, No. 2, 93-103 (2002; Zbl 1015.16007)] the author has found an action of the group of derived auto-equivalences fixing the trivial module on the cohomology of a given finite group. In the present paper he restricts attention to the subgroup of splendid auto-equivalences which are associated with splendid tilting complexes in the sense of Rickard. This restriction allows to use the Brauer construction. The action of this subgroup is shown to be compatible with the Mackey functor properties of group cohomology, that is, it behaves well under restriction, transfer and conjugation and also with respect to the local structure.NEWLINENEWLINENEWLINEIn a subsequent paper [Trans. Am. Math. Soc. 354, No. 7, 2707-2724 (2002; Zbl 1015.16006)], the author also proves compatibility with the action of the Steenrod algebra.